6.1 Areas Between Curves/27

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \color{red}\mathbf{y= \frac{1}{x} } & \color{green}\mathbf{y=x} \\ & \color{blue}\mathbf{y= \frac{1}{4} x } \\ & x>0 \\ \end{align} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{1} \left(x-\frac{1}{4}x)\right|dx + \int_{1}^{2} \left|(\frac{1}{x}-\frac{1}{4}x)\right|dx = \int_{0}^{1} \left|(\frac{3}{4}x)\right|dx + \int_{1}^{2} \left|(\frac{1}{x})\right|dx - \int_{1}^{2} \left|(\frac{1}{4}x)\right|dx = }

${\begin{aligned}&=\left[{\frac {3x^{2}}{8}}\right]{\Bigg |}_{0}^{1}+\left[\ln(x)\right]{\Bigg |}_{1}^{2}-\left[{\frac {1}{8}}x^{2}\right]{\Bigg |}_{1}^{2}=&=\left[{\frac {3(1)^{2}}{8}}\right]+\left[ln(2)-ln(1)\right]-\left[{\frac {1}{8}}(2)^{2}-{\frac {1}{8}}(1)^{2}\right]\\[2ex]&=\left[{\frac {3}{8}}\right]+\left[ln(2)\right]-\left[{\frac {3}{8}}\right]&=ln(2)\end{aligned}}$